Linear independence and Basis vectors

Linearly dependent

A set of vectors is set to be linearly dependent if at least one of the vectors in the set can be defined as a linear combination of the others.

Linearly dependent vectors may either overlap with each other or define the same plane. Such extra vector is not necessary to define a plane or space. Below is a relation between vectors u, v and w and scalars.

u = av + bw (3D space) w = av (2D space)

Linearly independent

If no vector in the set can be written as a linear combination of the others, the vectors are said to be linearly independent. Below is a relation between vectors u, v and w and scalars.

u != av + bw (3D space) w != av (2D space)

Two vectors u and v are linearly independent if the only numbers a and b satisfying au+bv = 0 are [a = b = 0] (known as trivial solution)

Basis vectors

Basis vectors are lineary independent vectors having the property that every vector in the space can be written uniquely as a linear combination of them. In other words, we can be scale basis vectors using scalars to reach all possible vectors in the given space.

Link: MIT Courseware 3Blue1Brown